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В рамках евклідової квантової теорії поля запропоновано метод вивчення характеристичних функцій для евклідових функцій Гріна в моделі з експоненціальним лагранжіаном взаємодії в двовимірному просторі при різних значеннях допоміжної константи зв’язку α і головної константи зв’язку g . Доведено, що в двовимірному випадку і α 2 > 8π відповідна евклідова теорія є тривіальною, а у випадку α 2 < 4π і достатньо великих значеннях g > 0 ця теорія нетривіальна при усунутих об’ємному і ультрафіолетовому обрізаннях. Виконання аксіом Остервальдера – Шрадера для евклідових функцій Гріна гарантує існування відповідної квантової теорії поля в аксіоматиці Гордінга – Вайтмана.
New method of investigation of the Euclidean Green functions for the nonpolynomial models of the quantum theory of field is proposed. The model with exponential interaction in the two-dimensional case is analysed. The monography by B.Simon emphathises the importance of these models from the scientific and methodical point of view. It should be noted, that at first the Wightman functions were used to state this theory. But after K.Osterwalder had found the connection between the Wightman and the Euclidean Green functions and the Nelson theorem of reconstruction had been proved, the construction of the relevant Euclidean model of the quantum theory of the field was reduced to the application of the theory of relativity methods (theory of Gaussian processes). While constructive building of the Euclidean Green functions the notions of triviality and non-triviality of the quantum theory of the field under removed ultra-violet and volume cut-offs, are of importance. The following approach to the solving of this problem is proposed. Characteristic measure functions, absolutely continuous relatively Gaussian measure are analysed. These functions are initiative ones for the Euclidean Green functions, that is why constructive building of the characteristic functions, in fact, provides the building of the mathematical model of the Euclidean theory of the field. As in the Lagrangian interaction two constants, principle and auxiluary, are available, the investigation α 2 > 8π and α 2 < 4π . 2 It has been found, that in the case α > 8π under removed ultra-violet cut-off the theory of the field is is carried out separately for the case trivial, that is, the Euclidean Green functions coinside with the relevant functions of the free quantum theory of 2 the field. In the case α < 4π the Euclidean theory of the field for the exponential interaction is non-trivial. In this case the infinite system of the integral equations, which is solved in the correspondingly chosen Banach space, has been obtained for the characteristic functions. The operator norm, which operates in this Banach space, was found to be less than 1. Then, basing on the Banach theorem, it can be stated, that the only solution of the equations system is available, and because of this volume and ultra-violet cut-offs can be removed. As a result, the limitation in the form of unequality on the principle coupling constant, has been obtained. |
